Function composition of symmetries -n, 1/n for FLT

Subj:  Re: Non-associative and non-distributive operations
Date:  19 Oct 2002 (sci.math)
From:  Nico Benschop
Orgn:  Digital Research : Finite & Associative
nhp wrote:
> Could somebody give an example of some "reasonable" operation
> a) not obeying the associative law (i.e. (x*y)*z != x*(y*z))
> b) not obeying the distributive law (i.e. x*(y+z) != x*y + x*z)
> c) not obeying neither associative nor distributive law
> Of course one can always introduce some quite arbitrary * and + such
> that the laws are not obeyed, but I'm looking for some operations
> which are actually used in maths or at least are somehow "reasonable".
> As simple as the laws are, they are (or maybe just because of that) at
> the heart of the maths. I think knowing some examples about what the
> laws exlude would give further insight into what these laws and
> calculus is all about.

The distributive law holds for one operation (*) over another (+)
if (*) is defined as repetition of (+). Think of the normal
arithmetic operations (^) over (*), and (*) over (+) for integers.

To get a feeling for how restrictive 'associative' property is,
notice that it is necessary for the important concept of 'iteration':
e.g.: (aa)a = a(aa) is denoted a^3 only in this associative case.

Imagine an order n closure of set S under some operation (.) defined
by its n x n composition table. There are n^{n^2} ways to fill such
table, of which n!  row/col permutations yield isomorphism's.

Now try to find the % of such square tables (closures) that are in fact
associative. Then you'll find for not too large n already (say n=10)
an almost negligible fraction is indeed associative. Meaning that
the 'associative' assumption for an operation: semigroup S(.) is
a VERY strong restriction (something like 'linear' for networks),
implying a lot of 'structure'... For non-associative "de beer is los"
(anything goes;-) - one example is exponentiation (^) over (+),
such as FLT (additive statement with exponential terms):

-- NB -
            (on semigroups & function composition in arithmetic)

Function composition of symmetries -n, 1/n for FLT

(news:sci.math . . 4apr98)

... on Number Theory and Arithmetic --- : in arithmetic ring Z(+, .) both (+) and (.) are associative and commutative. However, function composition ( associative, and not nec. commutative) is required to prove the general " triplet " structure of arithmetic Z(+,.) mod p^k, which is [ not only for p-th power residues, but also for each residue in units group G ] :

a+1 = -1/b, . . b+1 = -1/c, . . c+1 = -1/a, . . where : abc = 1 (all mod p^k)
in other words:

(a+1)b = (b+1)c = (c+1)a = -1 . . . with : abc = 1 . . . (all mod p^k)

See On discovering the triplets (mod p^k) . . . (double-click)

This triplet structure is used in a direct proof of FLT (Fermat's Last Theorem),
published in Acta Mathematica Univ. Bratislava (Nov.2005): Full text (16 pgs) :
. . . "Additive structure of the Group of Units mod p^k ,
. . . with Core and Carry concepts for extension to Integers.

The smallest example is Z (mod 3^2) = 2* = {2 4 8 7 5 1} = {2 4 -1 -2 -4 1} mod 9.

1+1 = 2 = -1/4, . . 4+1 = 5 = -1/7, . . 7+1 = 8 = -1/1 , with 1.4.7=1 mod 9.

This 3-loop holds for each n<>-1 in G mod p^k, for each prime p, and all k>0, except for the cubic roots of 1 mod p^k (p=1 mod 6) where the looplength is one: a+1 = -1/a (a<>1), and in a special case where 4 divides p-1 (so p=1 mod 4). . . . Moreover, the Hensel lift (of extending a mod p^2 solution to mod p^k for any k>2) can be "broken" by this a+1= -1/a solution, because the Exponent p Distributes over a Sum: (a+1)^p = a^p + 1 = a + 1 (mod p^k, EDS property in Core . : . see my paper on " Triplets ..." , ref[1] on my homepage).

The clue, to prove no loop longer than 3 exists in ring Z(+, .) are its two symmetries (= automorphism of order 2), seen as functions, namely:

Complement function C(n) = -n for addition ( neutral element '0' )

and : Inverse function I(n) = 1/n for multiply ( neutral element '1' ).

Notice that the triplet structure is seen sequentially as repeating the function -1/(n+1), substituting it into itself ("reflection" as it were): you will find that this returns n after 3 steps.
"Simple comme bonjour", would Fermat have said... ... although for p-th powers the first triplet is at p=59, and arithmetic doodling mod 59^2 in those days with his PC (Pascal Calculator = "Pascaline") is a bit unlikely. -- Yet, we don't need p-th powers to see the triplet structure of arithmetic, as shown above! Moreover, for p=7 the cubic roots occur already, which he very well could have found (for inequality FLT case1 they should be the only solutions mod p^2, which they are NOT). Check out

And the same 3-loop holds for ALL four possible compositions of these three elementary arithmetic functions C(n), I(n) and the successor S(n)=n+1. Notice that I and C commute, so 4 rather than 3! = 6 such "dfs" functions arise (dual folded successor functions).

So FLT (certainly case1) is NOT a purely arithmetic problem,
but requires more powerful function composition (semigroups).

As it were .: You need a diamond (sgrp) to cut steel (arithm).

For instance the above function is SIC(n)= -1/(n+1) = SCI(n), where function composition is normally from left to right -- following this nice notation advocated by Clifford/Preston in the reference work: "The Algebraic Theory of Semigroups" (AMS Survey #7, 1961)

You see that . : . looplength three = the number of symmetries + 1 = the number of operations + 1.

( . . I think this is not a coincidence, but a necessity . . )

The extension to inequality for integers follows from a variant of the Exponent p Distributes over a Sum (EDS) property of this solution of FLT case1 in residues ("two terms of a triplet are in Core" -- see my homepage ref[1] on the general Triplet structure of Arithmetic mod p^k, not only for p-th power residues).

Constructive comment is welcome.

------------- SCI = SIC ---------------sci.math--(23jul98)-----------
--------( Do not associate this title with Science = Sick ;-)---------
--------( Rather: an exercise in function composition )---------

Consider in arithmetic ring Z(+, .) the two basic symmetries -n and 1/n

as functions, in fact automorphisms of order 2, of Z(+) resp. Z(.):

Complement C(n) = -n, . . . Inverse I(n) = 1/n . . . . . . (n != 0)

And denote the successor function as: S(n) = n+1.

Notice that IC = CI (commute), but S does not commute with I or C. (so indeed SCI = SIC ;-)

Then compose all three functions in all possible (four) ways:

n(ICS) = 1- 1/n

n(SCI) = -1/(1+n)

n(CSI) = 1/(1-n)

n(ISC) = -(1+ 1/n)

Watch notation: function composition is from left to right.

Call such composition a "dfs" (dual folded successor) function.

  • Theorem (basic 3-loop of arithmetic):
    . . . . The third iteration of each dfs function is the identity function E
    . . . . . . ( nE = n, for all n<>0,-1,1 ).

    Proof: . . Let F_i be the i-th iteration of a function F. Then show: (dfs)_3 = E.
    . . . . By complete inspection the theorem follows easily.
    . . . . For instance n(SCI)_3: substitute -1/(1+n) for n three times:

    . . . n(SCI)_2 = -1/[1 -1/(1+n)] = -(1+n)/[1+n -1] = -(1+n)/n = -1/n -1

    . . . n(SCI)_3 = (1+n) -1 = n . . . QED.

    Similarly, verify the other three dfs functions to have period 3.

    An interesting consequence of this "3-loop" property of arithmetic is that for residues mod p^k (prime p>2, k>1):
    . . In units group G_k , of order (p-1).p^{k-1} , with all residues coprime to p :

  • . . Each 'a' generates a 3-loop of three inverse pairs, called "triplet":

    . . . . a+1 = -1/b --> b = -1/(a+1)
    . . . . b+1 = -1/c --> c = -1/(b+1)
    . . . . c+1 = -1/a --> a = -1/(c+1) . . . with abc=1 . . (all mod p^k)

    (provided division by zero is avoided, so e.g: a+1 <>0, etc.)

    Basic triplet example is in G(.) mod 9 = 2* = {2, 4, 8, 7, 5, 1} . . where 8= -1, 7= -2, 5= -4 (mod 9).
    . . Iteration ( start value '1' ) :

    . . . . . 1(SCI)_*: -1/(1+1)=4, -1/(4+1)=7, -1/(7+1)=1, with abc=1.4.7=1 mod 9

    Note : maximal period=3 is the number of symmetries +1 (coincidence?-)

    . . . For this function composition result applied to a proof of FLTcase1,
    . . . . see paper ref[1]in .dvi on my homepage; short intro ref[3] in .htm

  • Re: proof of Goldbach's Conjecture . . . sci.math-9sep99
    Subject:      Re: proof of Goldbach's Conjecture
    Author:       Nico Benschop
    Date:         9 Sep 99 07:04:03 -0400 (EDT) wrote:
    > Why you guys are wasting the time on this problem, I would never
    > believe anyone in this planet could prove the Goldbach's Conjecture
    > with 6-page, not even with 60-page. Much harder, guys. Handwaving
    > stuff won't help and old tools with which people proved the cases of
    > (1+n) (n>1) couldn't be used for case (1+1).
    > I truely believe we need  brand-new tool to task it.  ...[*]
    > It might take a couple of decades and even centuries. ...[&]
    Re[*]: Not quite: rather USE tools that are wellknown,
        such as the algebra of function composition
              [ associative, but not commutative:  f(g(x)) =/= g(f(x) ]
        to solve hard problems in arithmetic.
        (like on powersums: Fermat, Waring; or primesums: Goldbach;-)
        For instance: the two symmetries of arithmetic: complement -n
        under (+) about "0", and inverse 1/n under (.) about "1",
        have a fascinating 3-loop property that only can be seen under
        function composition. Namely, call them 'C' and 'I' respectively,
        and let 'S' be the Peano successor function n --> n+1.
    Now consider function SCI (from left to right apply S first, then C
    and lastly I), then you'll easily verify that this function 3 times
    applied in iteration, yields the identity function  E: n --> n.
    (do not divide by 0, so some restrictions hold: n<>0, and n<>-1 ):
    So:        n(SCI)= -1/(n+1)          applied twice more to itself:
           n(SCI SCI)= -1/{-1/(n+1) +1} = -(n+1)/{-1 + n+1} = -(n+1)/n
       n(SCI SCI SCI)= -1/{-(n+1)/n +1} = -n/{-n-1 +n} = n.
    Funny, magic, how come: a basic 3-loop linking the fundamental two
    symmetries of arithmetic and the Peano successor function ...!?
    E.g: this is the clue to all solutions of x^p + y^p = z^p mod p^k,
     namely the "Triplet": a+1=-1/b, b+1=-1/c, c+1=-1/a, with abc=1 mod p^k
     basic 3-loop structure of residue arithmetic mod p^k (prime p>2),
     (for ALL residues coprime to p) & a 'sideline' to integers: FLTcase1,
      breaking the Hensel-lift by taking into account the 'carry':
      the p-th power of a k_digit number (base p) has upto kp digits -
      ------- the carry makes the difference, for FLT ;-)
    Indeed: an OLD and well known tool (function composition) applied
    in a NEW way! -- That's why I find: mathematicians are sitting on a
    goldmine (semigroups = associative algebra, especially in_the_finite)
    without really knowing it... (like Shannon in 1938 suggested to apply
    Boolean Algebra - a commutative & idempotent form of arithmetic -
    to the specification and design of combinational logic circuits:
    Boole's work was some 90 years old 1848: his monograph precursor
    of "The Laws of Thought" 1854). For sequential logic synthesis (FSM:
    Finite State Machines = computers) no such fundamental & practical
    tool has as yet been developped, although its basis: semigroup algebra
    (=function composition) is already existant since 1928 (Shushkewitch).
    Re[&]:   It need not necessarily take that long: IF we, open_minded
        engineers, scientists, and yes: some mathematicians - (although
        the latter have the disadvantage of the specialist/expert:
        single focus;-) - USE the tools already developed by previous
        generations,  ...the older & simpler the better...
    Number theory without fundamental use of function composition algebra
    [semigroups, like Z(.) analysed additively, including non-cmt, finite,
    with divisors of zero, ..&c] I would dare to call not quite complete,
    missing an essential concept.
    ========= It takes Steel to cut Wood,
              It takes Diamond to cut Steel... ========================
    where:   Wood = All Practical Purposes (APP in Science & Engineering)
            Steel = Arithmetic, Calculus, Set_theory (classical methods)
          Diamond = Associative Function Compostion (Semigroups)
    Ciao, Nico Benschop -

    Subject:  Re: logic, combinations, permutations
       Date:  Thu, 3 Aug 2000 12:48:21 GMT
       From:  Nico Benschop
        Org:  Research
    Newsgrp:  sci.math
    Niek Sprakel wrote:
    > Is there any coherent, consistent and complete theory which relates
    >        permutations, combinations and logic?
    > I reckon permutations are embeded among multinomialcoefficients and
    > propositional logic is closely related to binomialcoefficients.
    A good context for these algebra's (of permutations, transformations,
    sets, arithmetic, combinational- and sequential logic, FSM: state-
    machines) is given by the common property of the corresponding
    operations: associative.
    Hence: semigroups (= associative algebra of functions)
           is their common context.
    Ciao, Nico Benschop --
     simple sgrps as FSM:
     integer state machines
    -- N.F.Benschop -- July 1998 --